Question

В окружности с центром в точке О проведена хорда АВ.Центральный угол АОВ равен 105 градусам.Через точки А и В проведены касательные к окружности, пересекающиеся в точке Р.Найдите градусную меру угла АРВ.

Answer 1

1. угол OAD = угол ОВD = 90 градусов - т.к. ОА и ОВ - радиусы, проведенные к касательным
2. Сумма углов в четырехугольнике равна 360 градусов. Значит в четырехугольнике ВОАD:
угол В + угол О + угол А + угол D = 360
90 + 105 + 90 + угол D = 360
угол D = 360 - 285
угол D = 75 градусов
Ответ: 75 градусов

Answer 2

Problem Analysis

We are given a circle with center O and a chord AB. The central angle AOV is 105 degrees. Tangents are drawn through points A and B, intersecting at point R. We need to find the measure of angle ARV.

Solution

To find the measure of angle ARV, we need to determine the relationship between the angles formed by the tangents and the chord.

Let's analyze the given information step by step:

1. The central angle AOV is 105 degrees. This means that angle AOV is 105 degrees.

2. The tangents drawn through points A and B intersect at point R. This implies that angle ARO and angle BRO are both right angles (90 degrees).

3. Since angle ARO is a right angle, angle AROV is a straight angle (180 degrees).

4. Similarly, since angle BRO is a right angle, angle BROV is also a straight angle (180 degrees).

5. The sum of the angles in a triangle is 180 degrees. Therefore, angle ARV can be calculated as follows:

- Angle ARV = 180 degrees - angle AROV - angle BROV - Angle ARV = 180 degrees - 180 degrees - 180 degrees - Angle ARV = 0 degrees

Therefore, the measure of angle ARV is 0 degrees.

Conclusion

The measure of angle ARV is 0 degrees.

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